Properties

Label 930.2.a.n.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -1.00000 q^{45} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} -6.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} -2.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} -1.00000 q^{90} -8.00000 q^{91} +1.00000 q^{93} -12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 8.00000 1.29777
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 1.00000 0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 2.00000 0.218218
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) −12.0000 −1.23771
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) −4.00000 −0.392232
\(105\) −2.00000 −0.195180
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 2.00000 0.188982
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) −6.00000 −0.552345
\(119\) 12.0000 1.10004
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) −6.00000 −0.541002
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 4.00000 0.350823
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 2.00000 0.172774
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) −12.0000 −1.01058
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.00000 0.648886
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) −4.00000 −0.320256
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) −6.00000 −0.460179
\(171\) 8.00000 0.611775
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −8.00000 −0.592999
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 2.00000 0.145479
\(190\) −8.00000 −0.580381
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) 4.00000 0.286446
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.00000 0.141069
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) −10.0000 −0.696733
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 2.00000 0.135769
\(218\) 2.00000 0.135457
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) −4.00000 −0.268462
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −18.0000 −1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 8.00000 0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −4.00000 −0.261488
\(235\) 12.0000 0.782794
\(236\) −6.00000 −0.390567
\(237\) 8.00000 0.519656
\(238\) 12.0000 0.777844
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) −6.00000 −0.382546
\(247\) −32.0000 −2.03611
\(248\) 1.00000 0.0635001
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 8.00000 0.498058
\(259\) −8.00000 −0.497096
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 6.00000 0.363803
\(273\) −8.00000 −0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −4.00000 −0.239904
\(279\) 1.00000 0.0598684
\(280\) −2.00000 −0.119523
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −12.0000 −0.714590
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) −6.00000 −0.356034
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 8.00000 0.468165
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −3.00000 −0.174964
\(295\) 6.00000 0.349334
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) 8.00000 0.460348
\(303\) −6.00000 −0.344691
\(304\) 8.00000 0.458831
\(305\) −2.00000 −0.114520
\(306\) 6.00000 0.342997
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) −10.0000 −0.568880
\(310\) −1.00000 −0.0567962
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 14.0000 0.790066
\(315\) −2.00000 −0.112687
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 2.00000 0.110770
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000 0.658586
\(333\) −4.00000 −0.219199
\(334\) −24.0000 −1.31322
\(335\) −2.00000 −0.109272
\(336\) 2.00000 0.109109
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 3.00000 0.163178
\(339\) −18.0000 −0.977626
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) −20.0000 −1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 2.00000 0.106904
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −6.00000 −0.318896
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) 2.00000 0.105118
\(363\) −11.0000 −0.577350
\(364\) −8.00000 −0.419314
\(365\) −8.00000 −0.418739
\(366\) 2.00000 0.104542
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 4.00000 0.207950
\(371\) −12.0000 −0.623009
\(372\) 1.00000 0.0518476
\(373\) 38.0000 1.96757 0.983783 0.179364i \(-0.0574041\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) 32.0000 1.64373 0.821865 0.569683i \(-0.192934\pi\)
0.821865 + 0.569683i \(0.192934\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) −18.0000 −0.920960
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 8.00000 0.406663
\(388\) −10.0000 −0.507673
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −6.00000 −0.302660
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 8.00000 0.401004
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 2.00000 0.0997509
\(403\) −4.00000 −0.199254
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 6.00000 0.296319
\(411\) −6.00000 −0.295958
\(412\) −10.0000 −0.492665
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −4.00000 −0.196116
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 8.00000 0.389434
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) −6.00000 −0.290701
\(427\) 4.00000 0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −24.0000 −1.14156
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 18.0000 0.851371
\(448\) 2.00000 0.0944911
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 8.00000 0.375046
\(456\) 8.00000 0.374634
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −1.00000 −0.0463739
\(466\) 18.0000 0.833834
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) −4.00000 −0.184900
\(469\) 4.00000 0.184703
\(470\) 12.0000 0.553519
\(471\) 14.0000 0.645086
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 8.00000 0.367065
\(476\) 12.0000 0.550019
\(477\) −6.00000 −0.274721
\(478\) 12.0000 0.548867
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 16.0000 0.729537
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 2.00000 0.0905357
\(489\) 2.00000 0.0904431
\(490\) 3.00000 0.135526
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −12.0000 −0.538274
\(498\) 12.0000 0.537733
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 2.00000 0.0890871
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) −16.0000 −0.709885
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) −6.00000 −0.265684
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −30.0000 −1.32324
\(515\) 10.0000 0.440653
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) −6.00000 −0.262111
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −6.00000 −0.260378
\(532\) 16.0000 0.693688
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 32.0000 1.37452
\(543\) 2.00000 0.0858282
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) −8.00000 −0.342368
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 8.00000 0.339887
\(555\) 4.00000 0.169791
\(556\) −4.00000 −0.169638
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 1.00000 0.0423334
\(559\) −32.0000 −1.35346
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −12.0000 −0.505291
\(565\) 18.0000 0.757266
\(566\) −22.0000 −0.924729
\(567\) 2.00000 0.0839921
\(568\) −6.00000 −0.251754
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) −8.00000 −0.335083
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 19.0000 0.790296
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −10.0000 −0.414513
\(583\) 0 0
\(584\) 8.00000 0.331042
\(585\) 4.00000 0.165380
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 6.00000 0.247016
\(591\) 6.00000 0.246807
\(592\) −4.00000 −0.164399
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 18.0000 0.737309
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 1.00000 0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 16.0000 0.652111
\(603\) 2.00000 0.0814463
\(604\) 8.00000 0.325515
\(605\) 11.0000 0.447214
\(606\) −6.00000 −0.243733
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 48.0000 1.94187
\(612\) 6.00000 0.242536
\(613\) −40.0000 −1.61558 −0.807792 0.589467i \(-0.799338\pi\)
−0.807792 + 0.589467i \(0.799338\pi\)
\(614\) −34.0000 −1.37213
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −10.0000 −0.402259
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −24.0000 −0.956943
\(630\) −2.00000 −0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000 0.318223
\(633\) 8.00000 0.317971
\(634\) 6.00000 0.238290
\(635\) 16.0000 0.634941
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) −1.00000 −0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 48.0000 1.88853
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 2.00000 0.0783862
\(652\) 2.00000 0.0783260
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 2.00000 0.0782062
\(655\) 6.00000 0.234439
\(656\) −6.00000 −0.234261
\(657\) 8.00000 0.312110
\(658\) −24.0000 −0.935617
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −4.00000 −0.155464
\(663\) −24.0000 −0.932083
\(664\) 12.0000 0.465690
\(665\) −16.0000 −0.620453
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) −28.0000 −1.08254
\(670\) −2.00000 −0.0772667
\(671\) 0 0
\(672\) 2.00000 0.0771517
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 20.0000 0.770371
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −18.0000 −0.691286
\(679\) −20.0000 −0.767530
\(680\) −6.00000 −0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 8.00000 0.305888
\(685\) 6.00000 0.229248
\(686\) −20.0000 −0.763604
\(687\) 14.0000 0.534133
\(688\) 8.00000 0.304997
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) 2.00000 0.0755929
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −4.00000 −0.150970
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 30.0000 1.12906
\(707\) −12.0000 −0.451306
\(708\) −6.00000 −0.225494
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 6.00000 0.225176
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 30.0000 1.11959
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −20.0000 −0.744839
\(722\) 45.0000 1.67473
\(723\) 2.00000 0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) 48.0000 1.77534
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 20.0000 0.738213
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 4.00000 0.147043
\(741\) −32.0000 −1.17555
\(742\) −12.0000 −0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 1.00000 0.0366618
\(745\) −18.0000 −0.659469
\(746\) 38.0000 1.39128
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 2.00000 0.0727393
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 32.0000 1.16229
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −16.0000 −0.579619
\(763\) 4.00000 0.144810
\(764\) −18.0000 −0.651217
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 14.0000 0.503871
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 8.00000 0.287554
\(775\) 1.00000 0.0359211
\(776\) −10.0000 −0.358979
\(777\) −8.00000 −0.286998
\(778\) 24.0000 0.860442
\(779\) −48.0000 −1.71978
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −14.0000 −0.499681
\(786\) −6.00000 −0.214013
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 2.00000 0.0709773
\(795\) 6.00000 0.212798
\(796\) 8.00000 0.283552
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 16.0000 0.566394
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 36.0000 1.27120
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 36.0000 1.26569 0.632846 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 6.00000 0.210042
\(817\) 64.0000 2.23908
\(818\) 14.0000 0.489499
\(819\) −8.00000 −0.279543
\(820\) 6.00000 0.209529
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) −6.00000 −0.209274
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) −12.0000 −0.416526
\(831\) 8.00000 0.277517
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) −4.00000 −0.138509
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 30.0000 1.03633
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) 26.0000 0.896019
\(843\) −18.0000 −0.619953
\(844\) 8.00000 0.275371
\(845\) −3.00000 −0.103203
\(846\) −12.0000 −0.412568
\(847\) −22.0000 −0.755929
\(848\) −6.00000 −0.206041
\(849\) −22.0000 −0.755038
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) −6.00000 −0.205557
\(853\) 2.00000 0.0684787 0.0342393 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 4.00000 0.136877
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −8.00000 −0.272798
\(861\) −12.0000 −0.408959
\(862\) 6.00000 0.204361
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) −16.0000 −0.543702
\(867\) 19.0000 0.645274
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 8.00000 0.270295
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 8.00000 0.269987
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −3.00000 −0.101015
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −24.0000 −0.807207
\(885\) 6.00000 0.201688
\(886\) −36.0000 −1.20944
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −4.00000 −0.134231
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) −28.0000 −0.937509
\(893\) −96.0000 −3.21252
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) −18.0000 −0.598671
\(905\) −2.00000 −0.0664822
\(906\) 8.00000 0.265782
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) 8.00000 0.265197
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 8.00000 0.264616
\(915\) −2.00000 −0.0661180
\(916\) 14.0000 0.462573
\(917\) −12.0000 −0.396275
\(918\) 6.00000 0.198030
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) −34.0000 −1.12034
\(922\) −36.0000 −1.18560
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −16.0000 −0.525793
\(927\) −10.0000 −0.328443
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −1.00000 −0.0327913
\(931\) −24.0000 −0.786568
\(932\) 18.0000 0.589610
\(933\) −18.0000 −0.589294
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 4.00000 0.130605
\(939\) −16.0000 −0.522140
\(940\) 12.0000 0.391397
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 14.0000 0.456145
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 8.00000 0.259828
\(949\) −32.0000 −1.03876
\(950\) 8.00000 0.259554
\(951\) 6.00000 0.194563
\(952\) 12.0000 0.388922
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −6.00000 −0.194257
\(955\) 18.0000 0.582466
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) −12.0000 −0.387500
\(960\) −1.00000 −0.0322749
\(961\) 1.00000 0.0322581
\(962\) 16.0000 0.515861
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −11.0000 −0.353553
\(969\) 48.0000 1.54198
\(970\) 10.0000 0.321081
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000 0.0320750
\(973\) −8.00000 −0.256468
\(974\) −40.0000 −1.28168
\(975\) −4.00000 −0.128103
\(976\) 2.00000 0.0640184
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 2.00000 0.0639529
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) −32.0000 −1.01806
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 1.00000 0.0317500
\(993\) −4.00000 −0.126936
\(994\) −12.0000 −0.380617
\(995\) −8.00000 −0.253617
\(996\) 12.0000 0.380235
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −4.00000 −0.126618
\(999\) −4.00000 −0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 930.2.a.n.1.1 1
3.2 odd 2 2790.2.a.k.1.1 1
4.3 odd 2 7440.2.a.b.1.1 1
5.2 odd 4 4650.2.d.u.3349.2 2
5.3 odd 4 4650.2.d.u.3349.1 2
5.4 even 2 4650.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.n.1.1 1 1.1 even 1 trivial
2790.2.a.k.1.1 1 3.2 odd 2
4650.2.a.d.1.1 1 5.4 even 2
4650.2.d.u.3349.1 2 5.3 odd 4
4650.2.d.u.3349.2 2 5.2 odd 4
7440.2.a.b.1.1 1 4.3 odd 2